Textbook: Introduction | Mathematics for Grade 10 PDF Download

 Page 1


 chaPTer 1 INTRODUCTION 1
1 
introduction
In this chapter, you will:
•	 revise some concepts we use in algebra
•	 be reminded of some mathematical conventions
•	 acquaint yourself with your calculator, as well as use it as a 
learning tool
•	 revise base quantities that we use in science technology, 
and in our daily lives
•	 revise the order of operations
•	 revise properties of operations
•	 revise computational properties of integers
•	 revise the concept of equivalence
•	 simplify algebraic expressions
•	 model some situations
•	 talk about expressions, equations, and identities
2153 TechMaths Eng G10 LB.indb   1 2015/10/22   3:40 PM
Page 2


 chaPTer 1 INTRODUCTION 1
1 
introduction
In this chapter, you will:
•	 revise some concepts we use in algebra
•	 be reminded of some mathematical conventions
•	 acquaint yourself with your calculator, as well as use it as a 
learning tool
•	 revise base quantities that we use in science technology, 
and in our daily lives
•	 revise the order of operations
•	 revise properties of operations
•	 revise computational properties of integers
•	 revise the concept of equivalence
•	 simplify algebraic expressions
•	 model some situations
•	 talk about expressions, equations, and identities
2153 TechMaths Eng G10 LB.indb   1 2015/10/22   3:40 PM
2 Technical Ma TheMaTicS Grade 10
1.1 algebraic language
exercise
1 Do you remember what the following mathematical terms mean? Copy and complete the 
table below.
explanation/meaning example(s)
(a) Product
(b) Quotient
(c) Sum
(d) Difference
(e) Factor
(f) Additive inverse
(g) Multiplicative inverse
(h) Identity for addition
(i) Identity for multiplication
(j) Coefficient
(k) Solution
(l) Reciprocal
(m) Input variable
(n) Output variable
1.2 Some words we use in algebra
When we join numbers or add numbers to each other, such as 5x + 3x we call it addition.
Subtraction is the process of subtracting or deducting one number from another,  
e.g. 13y - 12y.
The abbreviated process of adding numbers to each other a certain number of times is called 
multiplication, such as 2 × 3y × 6x. 
When we use the process of dividing a number into parts, we really want to see how many 
times a number is contained in another. 
An expression with one term only, like 6x
2
, is a monomial.
An expression that is a sum of two terms like -3x + 7 is called a binomial.
An expression that is a sum of three terms like 100x
3
 + 45x
2
 - 50x, is called a trinomial. 
The symbol x is often used to represent the variable in an algebraic expression but other letter 
symbols may be used.
2153 TechMaths Eng G10 LB.indb   2 2015/10/22   3:40 PM
Page 3


 chaPTer 1 INTRODUCTION 1
1 
introduction
In this chapter, you will:
•	 revise some concepts we use in algebra
•	 be reminded of some mathematical conventions
•	 acquaint yourself with your calculator, as well as use it as a 
learning tool
•	 revise base quantities that we use in science technology, 
and in our daily lives
•	 revise the order of operations
•	 revise properties of operations
•	 revise computational properties of integers
•	 revise the concept of equivalence
•	 simplify algebraic expressions
•	 model some situations
•	 talk about expressions, equations, and identities
2153 TechMaths Eng G10 LB.indb   1 2015/10/22   3:40 PM
2 Technical Ma TheMaTicS Grade 10
1.1 algebraic language
exercise
1 Do you remember what the following mathematical terms mean? Copy and complete the 
table below.
explanation/meaning example(s)
(a) Product
(b) Quotient
(c) Sum
(d) Difference
(e) Factor
(f) Additive inverse
(g) Multiplicative inverse
(h) Identity for addition
(i) Identity for multiplication
(j) Coefficient
(k) Solution
(l) Reciprocal
(m) Input variable
(n) Output variable
1.2 Some words we use in algebra
When we join numbers or add numbers to each other, such as 5x + 3x we call it addition.
Subtraction is the process of subtracting or deducting one number from another,  
e.g. 13y - 12y.
The abbreviated process of adding numbers to each other a certain number of times is called 
multiplication, such as 2 × 3y × 6x. 
When we use the process of dividing a number into parts, we really want to see how many 
times a number is contained in another. 
An expression with one term only, like 6x
2
, is a monomial.
An expression that is a sum of two terms like -3x + 7 is called a binomial.
An expression that is a sum of three terms like 100x
3
 + 45x
2
 - 50x, is called a trinomial. 
The symbol x is often used to represent the variable in an algebraic expression but other letter 
symbols may be used.
2153 TechMaths Eng G10 LB.indb   2 2015/10/22   3:40 PM
 chaPTer 1 INTRODUCTION 3
In the monomial -15x
3
, -15 is the coefficient of x
3
.
In the binomial 17x - 3, and the binomial 33x
2
 + 1 4, the numbers -3 and 1 4 are called constants.
exercise
2  Complete the table, using the completed first row as an example.
expression
Type of 
expression
Symbol used to 
represent the variable
constant
coefficient 
of
(a) 3x
2
 - 7x + 9 Trinomial x 9 x is -7
(b) 5s
3
 - 11 s
3
 is 
(c) -1,2t + p t is
(d) 105k 0 k is
(e) 11 - p + p
3
P
3
 is
1.3 Some mathematical conventions and expressions
Mathematicians have agreed upon certain things that make mathematical work much easier if 
everyone adheres to these agreements.
•	 Multiplication sign: The multiplication sign is often omitted in algebraic expressions.  
5 × p is written as 5.p, as well as 5p, and we write -3(a + 4) instead of -3 × (a + 4).
•	 Writing a product with a number and a letter symbol: It is common practice to write 
a known number (constant) first in a product; we write 1 1a instead of a11.
•	 When the number before a letter symbol is 1: It is common practice to drop the 
coefficient 1 in a number such as 1a and instead just write a or in the case of -1a we just write -a.
1.4 Quantities
Engineers, surveyors, scientists, and ordinary citizens encounter and work with quantities of 
different kinds every day. A quantity is anything that we can measure or count. Your height is a 
quantity, we can measure it. If you are 1,8 m tall, 1,8 is a number and metre is a unit of measure.
exercise
3 Give your own examples of quantities and their units of measurement. You may want to 
organise your information in a table such as the one below.
Quantity Unit of measure
2153 TechMaths Eng G10 LB.indb   3 2015/10/22   3:40 PM
Page 4


 chaPTer 1 INTRODUCTION 1
1 
introduction
In this chapter, you will:
•	 revise some concepts we use in algebra
•	 be reminded of some mathematical conventions
•	 acquaint yourself with your calculator, as well as use it as a 
learning tool
•	 revise base quantities that we use in science technology, 
and in our daily lives
•	 revise the order of operations
•	 revise properties of operations
•	 revise computational properties of integers
•	 revise the concept of equivalence
•	 simplify algebraic expressions
•	 model some situations
•	 talk about expressions, equations, and identities
2153 TechMaths Eng G10 LB.indb   1 2015/10/22   3:40 PM
2 Technical Ma TheMaTicS Grade 10
1.1 algebraic language
exercise
1 Do you remember what the following mathematical terms mean? Copy and complete the 
table below.
explanation/meaning example(s)
(a) Product
(b) Quotient
(c) Sum
(d) Difference
(e) Factor
(f) Additive inverse
(g) Multiplicative inverse
(h) Identity for addition
(i) Identity for multiplication
(j) Coefficient
(k) Solution
(l) Reciprocal
(m) Input variable
(n) Output variable
1.2 Some words we use in algebra
When we join numbers or add numbers to each other, such as 5x + 3x we call it addition.
Subtraction is the process of subtracting or deducting one number from another,  
e.g. 13y - 12y.
The abbreviated process of adding numbers to each other a certain number of times is called 
multiplication, such as 2 × 3y × 6x. 
When we use the process of dividing a number into parts, we really want to see how many 
times a number is contained in another. 
An expression with one term only, like 6x
2
, is a monomial.
An expression that is a sum of two terms like -3x + 7 is called a binomial.
An expression that is a sum of three terms like 100x
3
 + 45x
2
 - 50x, is called a trinomial. 
The symbol x is often used to represent the variable in an algebraic expression but other letter 
symbols may be used.
2153 TechMaths Eng G10 LB.indb   2 2015/10/22   3:40 PM
 chaPTer 1 INTRODUCTION 3
In the monomial -15x
3
, -15 is the coefficient of x
3
.
In the binomial 17x - 3, and the binomial 33x
2
 + 1 4, the numbers -3 and 1 4 are called constants.
exercise
2  Complete the table, using the completed first row as an example.
expression
Type of 
expression
Symbol used to 
represent the variable
constant
coefficient 
of
(a) 3x
2
 - 7x + 9 Trinomial x 9 x is -7
(b) 5s
3
 - 11 s
3
 is 
(c) -1,2t + p t is
(d) 105k 0 k is
(e) 11 - p + p
3
P
3
 is
1.3 Some mathematical conventions and expressions
Mathematicians have agreed upon certain things that make mathematical work much easier if 
everyone adheres to these agreements.
•	 Multiplication sign: The multiplication sign is often omitted in algebraic expressions.  
5 × p is written as 5.p, as well as 5p, and we write -3(a + 4) instead of -3 × (a + 4).
•	 Writing a product with a number and a letter symbol: It is common practice to write 
a known number (constant) first in a product; we write 1 1a instead of a11.
•	 When the number before a letter symbol is 1: It is common practice to drop the 
coefficient 1 in a number such as 1a and instead just write a or in the case of -1a we just write -a.
1.4 Quantities
Engineers, surveyors, scientists, and ordinary citizens encounter and work with quantities of 
different kinds every day. A quantity is anything that we can measure or count. Your height is a 
quantity, we can measure it. If you are 1,8 m tall, 1,8 is a number and metre is a unit of measure.
exercise
3 Give your own examples of quantities and their units of measurement. You may want to 
organise your information in a table such as the one below.
Quantity Unit of measure
2153 TechMaths Eng G10 LB.indb   3 2015/10/22   3:40 PM
4 Technical Ma TheMaTicS Grade 10
There is an internationally agreed upon system of units known as the International System of 
Units, abbreviated as SI. There are seven base quantities on which the SI is founded. You have 
worked with some of these quantities in your studies already.
exercises
4  Copy and complete the table below.
Base quantity Si (international System of Units)base units
name Symbol
thermodynamic  temperature kelvin K
amount of substance mole mol
luminous intensity candela cd
length
mass
time
electric current
5  Name appropriate units for measuring:
(a) The amount of cement that a bag will hold.
(b) The amount of sand that can be transported in a bakkie.
(c) The height of a school building.
(d) The distance between Johannesburg and Cape Town.
(e) The capacity of petrol in a car.
6  Name an item that can be used to estimate the following metric units:
(a) a centimetre
(b) a metre
(c) a litre
(d) a kilometre
(e) a kilogram
2153 TechMaths Eng G10 LB.indb   4 2015/10/22   3:40 PM
Page 5


 chaPTer 1 INTRODUCTION 1
1 
introduction
In this chapter, you will:
•	 revise some concepts we use in algebra
•	 be reminded of some mathematical conventions
•	 acquaint yourself with your calculator, as well as use it as a 
learning tool
•	 revise base quantities that we use in science technology, 
and in our daily lives
•	 revise the order of operations
•	 revise properties of operations
•	 revise computational properties of integers
•	 revise the concept of equivalence
•	 simplify algebraic expressions
•	 model some situations
•	 talk about expressions, equations, and identities
2153 TechMaths Eng G10 LB.indb   1 2015/10/22   3:40 PM
2 Technical Ma TheMaTicS Grade 10
1.1 algebraic language
exercise
1 Do you remember what the following mathematical terms mean? Copy and complete the 
table below.
explanation/meaning example(s)
(a) Product
(b) Quotient
(c) Sum
(d) Difference
(e) Factor
(f) Additive inverse
(g) Multiplicative inverse
(h) Identity for addition
(i) Identity for multiplication
(j) Coefficient
(k) Solution
(l) Reciprocal
(m) Input variable
(n) Output variable
1.2 Some words we use in algebra
When we join numbers or add numbers to each other, such as 5x + 3x we call it addition.
Subtraction is the process of subtracting or deducting one number from another,  
e.g. 13y - 12y.
The abbreviated process of adding numbers to each other a certain number of times is called 
multiplication, such as 2 × 3y × 6x. 
When we use the process of dividing a number into parts, we really want to see how many 
times a number is contained in another. 
An expression with one term only, like 6x
2
, is a monomial.
An expression that is a sum of two terms like -3x + 7 is called a binomial.
An expression that is a sum of three terms like 100x
3
 + 45x
2
 - 50x, is called a trinomial. 
The symbol x is often used to represent the variable in an algebraic expression but other letter 
symbols may be used.
2153 TechMaths Eng G10 LB.indb   2 2015/10/22   3:40 PM
 chaPTer 1 INTRODUCTION 3
In the monomial -15x
3
, -15 is the coefficient of x
3
.
In the binomial 17x - 3, and the binomial 33x
2
 + 1 4, the numbers -3 and 1 4 are called constants.
exercise
2  Complete the table, using the completed first row as an example.
expression
Type of 
expression
Symbol used to 
represent the variable
constant
coefficient 
of
(a) 3x
2
 - 7x + 9 Trinomial x 9 x is -7
(b) 5s
3
 - 11 s
3
 is 
(c) -1,2t + p t is
(d) 105k 0 k is
(e) 11 - p + p
3
P
3
 is
1.3 Some mathematical conventions and expressions
Mathematicians have agreed upon certain things that make mathematical work much easier if 
everyone adheres to these agreements.
•	 Multiplication sign: The multiplication sign is often omitted in algebraic expressions.  
5 × p is written as 5.p, as well as 5p, and we write -3(a + 4) instead of -3 × (a + 4).
•	 Writing a product with a number and a letter symbol: It is common practice to write 
a known number (constant) first in a product; we write 1 1a instead of a11.
•	 When the number before a letter symbol is 1: It is common practice to drop the 
coefficient 1 in a number such as 1a and instead just write a or in the case of -1a we just write -a.
1.4 Quantities
Engineers, surveyors, scientists, and ordinary citizens encounter and work with quantities of 
different kinds every day. A quantity is anything that we can measure or count. Your height is a 
quantity, we can measure it. If you are 1,8 m tall, 1,8 is a number and metre is a unit of measure.
exercise
3 Give your own examples of quantities and their units of measurement. You may want to 
organise your information in a table such as the one below.
Quantity Unit of measure
2153 TechMaths Eng G10 LB.indb   3 2015/10/22   3:40 PM
4 Technical Ma TheMaTicS Grade 10
There is an internationally agreed upon system of units known as the International System of 
Units, abbreviated as SI. There are seven base quantities on which the SI is founded. You have 
worked with some of these quantities in your studies already.
exercises
4  Copy and complete the table below.
Base quantity Si (international System of Units)base units
name Symbol
thermodynamic  temperature kelvin K
amount of substance mole mol
luminous intensity candela cd
length
mass
time
electric current
5  Name appropriate units for measuring:
(a) The amount of cement that a bag will hold.
(b) The amount of sand that can be transported in a bakkie.
(c) The height of a school building.
(d) The distance between Johannesburg and Cape Town.
(e) The capacity of petrol in a car.
6  Name an item that can be used to estimate the following metric units:
(a) a centimetre
(b) a metre
(c) a litre
(d) a kilometre
(e) a kilogram
2153 TechMaths Eng G10 LB.indb   4 2015/10/22   3:40 PM
 chaPTer 1 INTRODUCTION 5
1.5 relationship between quantities
exercise
7 Determine whether the pairs of quantities below are related to each other. If so, explain 
whether the value of the quantity given on the right increases or decreases as the value of 
the corresponding quantity on the left increases. 
(a) Number of minutes that have passed while 
driving a car at 120 kilometres per hour.
Amount of petrol in the petrol tank.
(b) Number of sweets eaten. Number of calories consumed.
(c) Perimeter of a square. Area of the square.
(d) Number of minutes that have passed while 
driving a car at 120 kilometres per hour.
Amount of petrol consumed by the car.
(e) The base of a triangle. The area of a triangle.
(f) A radius of the circle. The circumference of the circle.
1.6 Properties of operations
Operations with numbers have the following properties:
the distributive property
The distributive property simply means that if a number is broken into parts and each part 
is multiplied, the answer is the same as when the number is multiplied as a whole.
Example: 5 × 27 = 5 × 20 + 5 × 7 (We have broken 27 into 20 and 7.)
We can express the distributive property in a general mathematical way by saying that if a, b, 
and c are any three numbers, then a(b + c) = ab + ac.
We also say multiplication distributes over addition.
exercises
8 Calculate, without using a calculator, the value of the following. It is important to show 
your work.
 (a) 181,5 × 3 + 181,5 × 7 (b) 703 × 8 + 703 × 2 (c) 17 × 43 + 17 × 57
2153 TechMaths Eng G10 LB.indb   5 2015/10/22   3:40 PM